What is the remainder when $1492\cdot 1999$ is divided by $500$?
Solution: We have $1492 = 1500-8 \equiv -8\pmod{500}$ and $1999 = 2000-1\equiv -1\pmod{500}$.

Therefore, $1492\cdot 1999\equiv (-8)\cdot(-1) \equiv 8 \pmod{500}$. The remainder is $\boxed{8}$.